 So thank you very much for the invitation and the opportunity to present my work here today. And so I'm going to talk about two topics. One is explain briefly what this new large delimit of matrix models is and how it relates to SYK in particular. And the second is some applications of it and in particular to the study of phase diagrams of large N matrix quantum mechanics. So the talk really is in three parts. After general introduction, this largely in the phase diagrams. And the last part I will do most of it on the blackboard because I found it clearer to draw diagrams and graphs on the blackboard. All right. So let me just remind you very briefly the chain of developments over the last year also. So first of all, we found a class of models based on large N from unix systems with quench disorder, the so-called SYK models which were studied originally in the contents matter literature by Sashdev and George and Parcolet and so on and so forth. And these models very surprisingly turned out to display features that are expected for quantum black holes, in particular this chaotic behavior and the saturation of the bound but also the quasi-normal behavior and so on and so forth. So that was very interesting but also very surprising because clearly nobody at least no string theorist would have thought of looking at this sort of quench disorder models, fermionic quench disorder models to describe black holes. Then Witten proposed that, or remark, make the nice remark that actually there's another class of models which is going to display very similar physics as SYK just because the structure of the Feynman diagram is very similar. And these are the large N tensor models. And based on the tensor model technology developed by Burot and Rivasso and many other people, you can show that the leading large N diagrams have the same structure as in SYK. And so eventually you expect to find very similar physics. So this is sort of very interesting. The tensor models have the great advantage over quench disorder models in the sense that they are genuine quantum theories including at finite N. You don't have an averaging over Hamiltonian in these cases. You have a fixed Hamiltonian so it's really a genuine quantum theory at finite N and you can, in particular, study quantities that are not necessarily self-averaging and it makes sense. So that's very nice. However, both models, quench disorder or tensors, are quite exotic from the point of view of stream theory. It's clear that we've learned, especially via the open closed string duality, that the models, the quantum mechanical models that we expect to be good to describe quantum black holes are matrix models. And the reason for the fact that matrix models really are singled out in stream theories is very, very simple. It's just the idea that open strings have two endpoints and each of these endpoints carry shamp-a-ton factors which are essentially the two indices of matrices. So it's very, very difficult to find a similar interpretation for tensors or from three or higher and I have no idea how a tensor model could be embedded in any way in stream theory or make the link with standard holography which is based on using really open strings and D-brains, et cetera, et cetera. So it's really matrix models that one would like to study in my opinion and at least before this SYK era that's what people have been trying to do for many, many years in spite of the lack of very precise analytical tools to study the models in the regime one would like to consider. So let me be a little bit more precise about the kind of models that stream theory have been singling out. So these are models essentially that originates from D-brain constructions and all the precise stream theory constructions starts from a picture like this where you have a set of D-brains which eventually will be replaced by some closed string background in the large end limit. And the theory that lives on the brains is a matrix theory just because of what I said in the previous transparency because open strings attached to the D-brains carry these two indices at their end points. So they must be studied in the large end limit and is there a pointer actually or this is a laser, I guess this is a laser, good. So they need to be studied in the large end limit first of all and the basic degrees of freedom that you always find in these theories this is the basic ingredient that you have in all models they are bosonic matrices xij which carry an additional index mu corresponding to the motion of the brains transverse to their world volume. Okay so this is an ingredient that you have in absolutely every brain construction so this is a very important ingredient that you need to have in all these models. Good. So now this being said, I felt this very brief introduction let me talk about something that looks very very different very unrelated but to me was very important development at least at the level of intuition is the work by M. Paranetal which was also followed by series of interesting papers by Minoalan collaborators where these people have studied the large space-time dimension limit of classical general relativity. And this is much less trivial than what might have thought at the beginning because it turns out that there is of course a huge simplification in this limit which is due to the fact that the gravitational law is a power law and the power is proportional is essentially linear in the number of space-time dimension which means that the gravitational field decays very very quickly in large dimensions. So when you have a black hole, a very large number of dimensions all gravitational effects are localized very near the horizon. This of course simplifies dramatically the analysis because you can zoom into this region but in spite of the simplification, this authors shows that you keep all the fundamental non-trivial features of black holes. For example, you can compute the quasi-normal spectrum in the large D expansion and the results are non-trivial. You can study non-linear effects too like you can work out how the collision of black holes go in this approximation and all this is very non-trivial and described quite accurately in this approximation. So that's a very interesting idea that large D, large number of space-time dimension in general relativity, is simplifying a lot but still keeps the underlying physics untouched apparently. You can still describe what you want. And to me, it provided the motivation to study the large D limit of matrix quantum mechanics where now what is this capital D? Well, it would be from the point of view of the brain constructions the number of dimensions that are transverse to the brains from which you built your holographic correspondence. So you have capital D, so the number of dimensions of the brain world volume will be kept fixed. It could be one, two, three, this is fixed but the number of transverse dimensions is capital D and this will be taken to be very large. So we've seen before that the models contain matrices X which are labeled by this index Q that goes from 1 to D so at the end of the day everything boils down to studying models of this sort. So you have a Lagrangian with a kinetic term here I am looking at the example of quantum mechanics but it could be a field theory, I would replace this kinetic term with some wave operator that doesn't change the discussion and you have a bunch of interaction terms and the interaction terms here I'm writing them in a very very generic way. There are traces of a bunch of X's and of course the mu's must be paired 2 by 2 in order for the model to be invariant under the rotation group transverse to the brain. So there is a rule here to pair the indices mu one with another one etc etc and that defines a particular interaction that I write like I B. So this is very very generic, this is a very large class of matrix models that you might want to study. Now you want to go to the large n limit and the usual large n limit of course is defined by keeping the couplings Tb here fixed. This is the so-called Toft scaling. So this is true because I've been careful to normalize the Lagrangian with a factor of n in front so the usual Toft scaling with this normalization is indeed that the Tb are kept fixed. So that's well known, that's the large n limit that we want to take in any case we want to go to the planar limit. What about large d? Well the large d limit is, there is one simple way to consider it. It's just by using the intuition we have from vector models. So vector models have been studied for many many years, I think the techniques to solve them have been known in the 80s already and so you can of course when you have matrices with this additional index mu you can consider them as being vectors built of matrices and you can say okay now I'm going to some sort of mixed limit both the planar limit of matrix model and some large d limit of vector model. Okay it's not difficult and doing this is simply taking the Tb fixed at large d2. So the large the standard vector model large d limit the scaling if you put the d in front here is Tb fixed. Okay so large n, a lot of this Tb fixed, large d standard vector model like is also Tb fixed and not surprisingly when you do that it's an interesting approximation to consider you get an expansion for example of the free energy of this kind so you have a double expansion the expansion in powers n to the 2 minus 2g where g is the usual genus of the Feynman diagram and then you also have a sum over inverse powers of d d to the 1 minus l that corresponds to the large d expansion nothing here is surprising. So the mu's are pairwise contract. Yes absolutely so the particular pairwise contraction that you consider would define a particular interaction okay I didn't indicate it here you have many possibilities and one gives you a particular term. And the scaling does not depend on the order of the model. Sorry I didn't. The scaling does not depend on the order of the monomial because you expect the more contraction you have the more. So in the vector model like scaling everything is very simple you just have this d in front and this is an odd invariant term so d does not enter here anymore. So that's it. S I think the question was about the order. So S is anything. S is the order of the interaction. That's I think. Okay that's okay you can include interactions you know S equal to is the quartic term you can have six ticked terms anything you know this is completely general here. Sorry. What is the interpretation of l? l so l counts in the expansion at large d. Okay so you have a double expansion the genus counts at large n l at large d. You have also an interpretation for l it's the number of loops of bubbles for those who knows how the Feynman graph expansion of vector models goes you have a dual the drawing of the diagrams in terms of loops of bubbles and l counts the loops of bubbles. Okay so these are all well known. Fine so it looks like we have some nice approximation to matrix models by mixing up these two ideas of large n matrix models and large d vector models. However this is a very boring maybe not so surprisingly. This when you do this essentially what you get is vector model physics and one intuition that you can have that allows you to understand this is because the large n and the large d limits when you define them in this way actually commute. So you can take first the large n limit let's say restrict yourself to planar diagrams and then take large d which reduces even more the number of planar diagrams that you have considering or you could do the other way around. Fix n take first large d then you have these trees of bubbles if you like at leading order and then if you like take the large n to restrict to planar trees of bubbles but of course since you can take n first sorry if you can since you can take d first to infinity this is done the usual vector model like way and the large d model at fixed n is essentially a model with a very large number of vectors if you like each element of the matrices are vectors but apart from that you don't get anything really interesting it's a normal vector model physics with additional decorations if you like due to the indices. How do you contract these indices? You use the metric right? No no no so oops sorry no so here you have an odd invariant quantum mechanics or model. Yeah but how do you contract the indices? The space time indices. So the mu's it's odd invariant so so it's just delta function I mean it's a you could use the metric the metric which is delta delta this is quadratic plus multi-trace right in the large dimension that's why it's not multi-trace so here it could be multi-trace but here I have been looking only at single trace interactions. No no no this is a trace in n but from the point of your large d it's multi-trace. That's why it was confused by the fact that the scaling is always the same. Yeah that is e you always you always couple two indices together. So let me write down sorry let me write on there is really nothing here so let me write down an example two examples. Every sum of d is like a trace in d. If you like yeah so you could have this is one interaction okay so if n equal one let's say you go to the vector model limit that would be the completely standout x to the fourth vector model this is the first probably one that people have studied this od vector model okay n equal one this gives that now you have another possibility because n will not be one for me which would be a contraction of this. I'm saying from the point of your d all of them are all these are all multi-trace. If you like that's that's always the case I mean that that's you know this is x4 you know if you have one vector the interaction is x to the fourth x to the sixth so you could call that multi-trace because it's x squared. You can use other higher tensors and then you will have no trigger interactions in the d direction. No so we can discuss but no I think there's nothing here that you can do you cannot do anything and if you want od invariance that's the most general thing you can you can do all right fine so this is a little bit disappointing of course maybe not surprising because it's not easy to find a good approximation to the sum of planar diagrams but still I had this idea of n-paran in my head so I sort of believed that something could work here I don't know why this is doing this way all right so how to go around this problem and the idea is actually as follows the problem we have is clearly that the large d limit defined in the ordinary way do not include enough diagrams okay you get rid of too many diagrams at leading order so maybe an idea is to improve the situation by enhancing enhancing some of the couplings in the large d limit that is to say instead of taking large d at fixed tb let's now consider a scaling of this sort where tb is d to sum power g of b where g of b will be some number which can be strictly positive times lambda b and let's continue consider the limit of large d at lambda b fixed okay that's maybe naive on the first step but that's sort of natural I want to keep more diagrams so that the limit is more interesting so this is clearly a way to do it fine naively of course if you sit down and think about it a few seconds it looks impossible it looks very naive indeed if one enhance a coupling in a scaling like this diagrams that were that contain a very large number of vertices of this of the of the enhanced type which had some particular power of d in the standard scaling we now have a new power of d which can be arbitrary large and if you consider diagrams with you know an infinite at the end of the day number of these insertions they will blow blow up and so the largely limit does not exist so that's that's that's a very obvious remark that you can make which means that the tough scaling really is very delicate you know the typically with the tough scaling if you enhance couplings the limit does not exist anymore and if you diminish the scaling then the limit becomes completely trivial so it's dedicated to have a scaling where you actually keep enough diagrams to make an interesting limit and it looks like we are very constrained here however and that's so the main point of the first part of my talk that's the main result I want to point out the fact is that form vector matrix model there's a remarkable feature that holds and that saves the day it turns out that the powers of d and of n in a given diagram are not independent they're related so that's the completely new result I think that was absolutely not realized before that there's an interrelationship between n and d let me give you a rough intuition of why this could be the power of d in a diagram is related to the number of loops that you can made of the od lines you know the lines of od indices the more loops you have the more powers of d you have on the other hand the power of n is related to the genius of the surface on which you can draw the graph know it's clear that the number of loops that you can draw on a remand surface of genius G is sort of constrained by the genius so it's not so surprising that you might get a relation between the power of d and the power of n okay so that's at the level of intuition I'm going to make very precise statements very soon but first so let me give you the result in a more precise way so and I'm going to do that on examples for those of you who want the precise recipes come after the talk but on examples so here I've drawn I have I'm considering four three so a free typical interaction terms that you might want to contemplate so what is the precise rule to actually compute this number G of B which is called the genius of the interaction it has nothing to do with the genius of the final diagram so first of all for term of this sort it turns out that the genius of the interaction is zero okay boring so this I cannot do anything with with a with an interaction like that if I include it it will be like a vector model there's nothing you can do with it however with this guy it turns out that you can enhance the coupling with G of B equal one half so you can insert an additional square root of D in your Lagrangian in front of this term and it will still make sense in a way that I'm going to make more precise in a couple of minutes for this guy the genius would be one so you could put a factor of D enhancement in front of the company okay and here I've just drawn you know this would be the fat graph representation for a term like this this is the fat graph representation for this term you see that the odd loops the odd lines here do not cross and here's a cross so it's clear that the way you can draw them on the remand surface will be different and that's why you can have different rules here so now here is the precise result with this rule you can show that the highest power of D for planar diagrams is actually D so even though I have enhanced the coupling for example of this guy for remand for a planar diagram I still cannot possibly build diagrams that have a power greater than D what does that imply immediately for the old scaling for the old vector model like scaling an obvious consequence of this is that these vertices actually could not contribute in the planar limit if they could contribute in the planar limit then you could add them and have the limit blow away okay so so consequences that indeed they cannot contribute in the ordinary vector model limit but no they will contribute at leading order but the contribution is tamed more generally in models with un cross un symmetry so this is valid even for air mission models now in models where you would have a un cross un symmetry you can extend this result to any genus and the result is that the highest power of D of a diagram of genus G is D to the power one plus G so there is an upper bound on the power of D for fixed genus in other words the large D limit at fixed genus will exist okay so that's the main result technical result and now this is the new how the new expansion looks like you still have the usual large n expansion and at fixed genus now you can expand at large D with the new scaling so in the new scaling the expansion parameter is one over square root of D and the highest power at fixed G as I said is is one plus G and of course I'm sorry this n should be a D okay this is the large D expansion so that's a very bad T for okay so this here it's a D to the power one plus G minus L over 2 okay it's the large D expansion and G here is the old genus so G is the genus of the Riemann surface of the Feynman diagram the usual one so this this is correct everything is correct here except this n that should be a D sorry about this so Frank if you think of x new as a tensor do you get one or far as one? So ask again in two minutes after a couple more transparencies so what about taking n equal what about taking D n squared oh that wouldn't work that wouldn't work so the limit would not exist then if you take D equal n squared all right so crucial property of this new large D limit they do large D and large n now do not commute okay if you want to take this new large D at fixed n it's clear that you can consider diagrams of arbitrarily high genus and those can have arbitrarily high powers of D so the large D limit at fixed n do not exist but if you first take the large n limit let's say you restrict to planar diagrams this is the leading order then the new large D limit exists and you get a new one over square root of D expansion of Feynman diagrams so sorry okay so the result of all these tricks is that you get a new sort of approximation to the sum of a planar diagrams that is associated with large D but which contains much more diagrams than what you would get in a vector model like large D and what this new diagrams are well there are melons or generalized melons which makes the link with syk and tensor models okay so you're getting here again if you like it's a third way to get again the same sort of general structure of melons or generalized melons of Feynman diagram in a systematic expansion at large D for matrix models so the main bonus of course is that we are not dealing with completely standard matrix quantum mechanical or field theoretical models okay you don't need to go to exotic quench disorder or tensor models it's completely standard matrix models in a new systematic expansion at large D from that point of view it's not surprising that you would get black hole physics out of this set of Feynman diagrams since they are precisely coming in models coming in in models that string theory have been telling us for many years that they should be describing black holes there was a question yes so the Lagrangian okay I could come back the Lagrangian is a completely general od symmetric matrix model take anything and I gave you the rules not in details because I didn't really tell you how you compute this g of b but there's a rule the mathematical rule any model you give me any matrix model which is od invariant so you have matrices x mu maybe you have x mu y mu z mu okay of course I didn't say that but it's obvious you can have flavors of matrices maybe some of these matrices are bosonic others are fermionic that doesn't enter into these arguments anything so okay let me come back then okay here it is here it is okay so you have a kinetic term I wrote it for quantum mechanics you could do field theory of course I could write here a kinetic term for higher dimensional field theory and then any interaction i b of x and here I restricted myself to a single trace interaction I could actually also do the discussion for multitrace interaction that will be done in a paper that will appear very soon and why does that only have balance so in the large deal in the new large d limit that I have defined it turns out that the leading diagrams are melons or generalized melons why okay I can I can you can read the paper you can prove that okay so that's the result I don't I don't give the proof here but that's the result right they are allowed in three indices so they look like that they look like because you have these three indices but they're not it's both like tensors and with addition some additional ingredients but you can prove it okay that's the result that's the the new result and something confusing because in the tensor models d is equal to n is taken to infinity yes and they have the same symmetry you're saying that n has to be taken first yes yet you end up with the same diagram only at leading order only at leading order yes because the leading order for d equal n is matching here the leading order when you take first n to infinity and then d to infinity that's not hard to tell I can I can give you d that's not very hard to understand that's not surprising so you mentioned something about the higher dimensional theories yes how do you investigate if you can get any strong interacting we started it's sort of hard we're not there yet I will I don't know if I will have time but even for the quantum mechanical example I would like to mention that many things actually are still not understood so all right so let me try to maybe go a little bit faster okay this this was okay so the the last remark it was that you know I I think the reason to me why syk is relevant to black hole is really from what I just said that the same structure of diagrams can actually also show up in in this context of matrix quantum mechanics and then there's no surprise that this could describe black holes okay another bonus I can mention briefly is that it's possible to deal with the planar limit of Hermitian matrix models so where you just have one un symmetry not un cross un so that's very unlike tensor models where you need a un symmetry per index okay otherwise the large n limit does not work and this is something you can do precisely because you take first large n so you first go to planar and then large d so that's an additional sort of bonus compared to the usual tensor model even though a lot of the underlying techniques are similar but that's a new a new thing to my to my knowledge that's the only way to actually reduce the symmetries of these models and still having a limit that makes sense the new large d limit let me mention that very briefly is consistent with linearly realized supersymmetry so the large d scaling that you have to take actually commutes with supersymmetry so supersymmetry relates different interaction terms so you have to check that the way it relates this interaction terms is consistent with the rule to assign powers of d to the couplings and that's working that is okay so you can study also supersymmetric models in values number of dimensions zero plus one but also one plus one and two plus one sorry the twist says that you don't have arbitrary trace i guess you you must take the one which appeared in Jungian so four if you take no so up to four super charges i can have arbitrary traces because it's like a super potential yeah and so i think the most interesting models in this class are probably models with four super charges which are very hard to study with ordinary techniques but which are known to be actually the most interesting so you know it's sort of nice maybe that these are the one that are also natural here you know so so yeah so you can be much more general and i think they are the most interesting for the physics point of view so this is for the future but they are there okay just to a very briefly answer a question by Kostas with how is it related to gyro scaling etc so it turns out that this idea of enhancing coupling if you like came back to many people but in particular there's a very nice paper by Karossan Tanasa in 2016 which was then used by Klebanov and Tarponowski last December where also they used this idea of enhanced coupling in a very special instance in a very special case now just to mention it very briefly this is a work essentially that will be i think published soon done with Vincent Rivasso and Guillaume Vallette it turns out that the gyro scaling the standard bonzon gyro Rivasso scaling of general tensor models can be improved in the sense that you can enhance the coupling in a way which is like that so the new scaling the old scaling would be mua fixed now you're going to take lambda a fixed so all the couplings with this degree so-called gyro degree strictly positive will be infinitely enhanced there's a way to do that and what is quite nice is that the the new large and expansion of tensor models that you get in this way is the diagrams are no classified by a new quantity so the traditional quantity in tensor models is the the gyro degree i don't want to explain many details what it what this is but let me just tell you that the leading graph graphs of degree zero which are also called super planar intuitively there are graphs that are essentially defined by the property that whatever way you draw them on the surface they always planar which means that you know to draw a graph on a surface you need to pick at each vertex a cyclic ordering of the lines incident to the vertex so here you have many lines because you have all these tensor indices the gyro degree graph the degree zero graphs are those that whatever cyclic ordering you choose they always planar okay so this is the gyro degree now with the enhanced scaling large and scaling that we're considering the the gyro degree is replaced by what we call an index and the index zero graphs are different the index zero graphs are as follows when you have a tensor with our indices you can always pick two of them and forget about all the others that define a matrix model the graph associated with the tensor model can then be written as a standard fat graph just keeping you know the lines the strands associated with the two indices that you've picked for each such choice you just have a genus associated with the diagram no take any possibility pick any two indices you find a genus for each choice same over all this genera this is the index okay so it's very natural in some sense it's like testing all the possible ways to make matrix models from a tensor and you sum all the genera that result that's the index and so the graph that dominates of index zero are the graph for which the matrix model graph that are associated two are always planar whatever way you define the notion of matrix by picking up two indices all right so this will be published soon too all right in the last 20 do you have questions so this ends the first part of the diagrammatic large n large d part now i would like just to discuss very briefly the sum of the applications and in particular applications to the phase diagram of the models so we now have a vast class of models that you can that we can study and they are not exotic at all okay so the simplest one would be something very similar to syk or some generalization by sash death you consider a hamiltonian of this sort based on matrices psi mu which are direct fermions so they are really quantized by just deciding that they satisfy this standard anti commutation relations okay very basic and and very natural hamiltonian to look at the crucial new ingredient is that we're going to take large n so planar but also large d with the square root of d here that enhanced the interaction and capital lambda capital m will be the two parameters and they will be kept fixed okay so this is an example of a matrix quantum mechanics that you can study other example bosonic so you take a hamiltonian with bosons and an interaction similar to this one this is an unstable model it turns out you can check it's not too hard to check that an interaction trace of x mu x mu x mu is not stable you can find directions where the the term can be arbitrarily high arbitrarily low but you know that at large n and stable models still make sense so you can consider it you can also consider bosonic stable models for example this is a very nice interaction x mu x rho x mu x mu x rho x mu which is manifestly positive definite because essentially it's the square of her mission matrix here so this is very well and well defined so why don't you take the commutator squared in the second so commutator squared is not good because it will mix so that's an unfortunate situation it will mix x mu x mu x mu it will mix these two terms no sorry it's equivalent to the first one because you only keep the dominant one you understand no so the the second one it depends how then the problem is then that the large descaling of these two will not be the same so you cannot keep the commutator structure if i was very brief but if you remember this cannot be enhanced okay so the large descaling here will be standard vector model like this this one can be and should be enhanced if you don't enhance you get vector model physics not interesting but then if you enhance it they don't play the same role and they no longer appear as a commutator squared and of course you can do super symmetric models with two or four supercharges taking for example a super potential of this sort which no will of course yield a stable model if you take a super potential of this sort the potential will be actually this term and you will have additional fermionic terms so we've we are now studying all these guys and it turns out that very little is known about these models what is their phase diagram what are their properties six months ago i think nobody would have even dreamt of being able to say anything reliable about these guys in the strong coupling region okay so it's a completely new arena to play and even it turns out that even for the fermionic models that look very much like syk so the the first Hamiltonian i draw there are very big surprises and i hope i have time maybe yes to describe briefly what these are and so this is a work that is a non-near completion with tetsuo azeyanagi in brusseuse and fidelschapposnik in sel so let me just give you the intuition of the parameter space or the phase diagrams of these models what do you expect okay so we have three parameters of course you could have more more complicated guys but the the basic physics is captured with these models with three parameters so you have the mass okay which gives just the the frequencies of the harmonic oscillator limit of the model of the weekly coupled limit of the model you have a coupling lambda and the temperature all this free i choose to be of dimension of a mass okay or of an energy so really the dimensionless parameters you have only two and i will call them little m which is capital m divided by lambda little t capital t divided by lambda i use that because of course the model with capital lambda is zero is trivial it's just a harmonic oscillator okay so capital lambda will never be zero maybe you can set it to one that set the scales and then you have really two interesting parameters temperature and m so what is this plane what does it look like it has different regions first you have this region up here which is large t at fixed m okay and fixed m could be m equals zero if you take m equals zero this is actually syk so this regime is the perturbative regime where the Feynman graph that are used in syk to eventually go to strong coupling after summation are defined okay syk it's also defined perturbatively to start with because it's based on Feynman diagrams so the Feynman diagrams of syk the perturbation theory corresponds to this regime here but of course you have another regime perturbative regime which is actually more standard i would say which is the standard textbook perturbation theory of a model of very large m at fixed temperature so this would be the standard perturbation theory that you have in textbook around the fork vacuum so this is where you know the mass of the harmonic oscillator if you like term is very high the coupling is very small so you perturb with a standard propagator around the fork vacuum and and you do that and this would be this region here but you're sorry you know salate mass is the same for all the salators yes because of OD environment this is non-Kamma not Kalmogorov or another regime so it's not integrable and chaotic for any in classical dynamics it's chaotic for 50 it's chaotic yeah it's chaotic because come cannot be applied and for any small perturbation it is it is it is but it is perturbative in the sense that for example if you want it's not perturbed no in this regime yes if you want to compute the free energy for example okay as function of temperature you do a one-loop calculation you get an extremely precise formula for the free energy that's what i mean by perturbative but you're right if you look at the real-time two-point function even here very far here it will be chaotic so it will it will develop non-perturbative features that will not be captured by perturbation theory but it is perturbative in the sense that you have a very small coupling and in particular to compute any thermodynamic quantity entropy for energy etc i could reliably don't do it in a perturbative expansion even the three-level result would be already precise here that's what i mean but you're right and if i have time i'll come back on what you say do you respect to gay singles or sorry do you respect to gay singles so you could but at leading order that's something i didn't say you don't need because the gauge group rank is n since d is large at leading order all my actions are of order n squared d so a gaging un is a sub leading you know the gaging un will give you ghost terms etc but all these terms know only about the un part not the od part so they give contributions of order n squared which are sub leading in the large d expansion okay so that's a very good point temperature would be zero yes here yes in the large g limit but if you compute the one over d correction you'll start to see this okay it's like you know very well of course okay so you the effective action that you compute will be next n squared d but all the terms coming from the integral over the unitary group this trigonometric Vandermonde etc that will be of other n squared so sub leading at leading order but you could see the effect but that would be a harder calculation we started to think about it but i think it's for later because it's a sort of hard calculation all right so what what what did i say here i'm just saying that there's an effective coupling in this theory which is really the minimum of one over m and one over t okay the one over m is the usual maybe most familiar coupling that governs perturbation theory here but you also have one over t here and the reason that you have that is that of course with fermions the zero mode takes a mass which is proportional to the temperature because of that the large temperature limit for fermions is always weakly coupled and here you can start to see an important difference with bosonic system with bosonic system the large temperature limit is has no reason to be weakly coupled it's classical but it's not weakly coupled for fermionic system it will always be perturbative and that's at the basis actually of the whole syk business because that's how they define the diagrams here fine okay so this is the phase diagram perturbative region here but here a non-perturbative region now in this perturbative region of course you can compute the the entropy is very simple the here it's it would be non-zero because you perturb essentially around zero Hamiltonian so you perturb around the state of maximal entropy so just log two per degrees of freedom here it would be the most more standard perturbation theory where you have just harmonic oscillators so here the the the the entropy would go to zero exponentially fast when the temperature goes to zero as usual the fantastic thing is that in these models you can actually compute the entropy all the way down here to zero temperature which is very strongly coupled so the strongly coupled so i don't explain how to do that because that's not syk you don't have auxiliary fields etc so there's a little bit of technology that i'm not talking about but still you can do things that are similar to syk okay i'm not explaining that but you can compute the entropy in the zero temperature limit little t equals zero that's the result the catalon number divided by five by pi plus one quarter of log two that's a number and numerically you can actually go from very weak coupling to super strong coupling that's a marvelous property of this of these models but you can do it everywhere okay so now this is the puzzle of the phase diagram here here it's not mysterious okay you start from the perturbative regime which already has a non-zero entropy at zero's order you're going to sum up planar diagrams for Feynman diagrams and okay you do that that's syk like things and you go to a non-zero zero temperature entropy with all the syk like physics that we are now starting to be familiar with however if you do it here that's completely different here this is harmonic oscillator physics with a zero entropy at zero temperature and actually exponentially vanishing entropy at zero temperature how can you go then in this direction you know syk is this that's well known well understood what about this what is this part of the phase diagram how do you make the link between ordinary perturbation theory harmonic oscillators and this regime of non-zero entropy at zero temperature okay let me skip that that that that's the what you would have for boson and if you want the answer i think i have i should have five minutes i can draw on the blackboard the answer okay to this to this puzzle so maybe if you can switch off the uh this and i will use i just need one blackboard and i will just give you the answer in this model but that's the basic physics that opens the way to understand the phase diagrams for many more models so you had here n and you had here t okay and i want to understand how i go there let me draw another curve which is the entropy is very useful as a function of mass and i will do that for various temperatures let me do it first for t equals zero so for t equals zero it's the well you have the well known syk like solution generalized to finite m syk would be m equals zero if you like but finally it's just a generalization that sash death actually has studied so that's well known you can draw and that's what you get so here you get this catalan this number is catalan over pi etc and and this is what you get here you get so you increase the mass the entropy diminishes but remarkably so that's one first new feature you realize that the solution ends here at some critical mass so the syk solution like solution actually does not exist for all values of n okay so this is the first crucial feature people have not realized that in the condense matter literature for one reason it's because they studied the model not as a function of the mass but as a function of the charge okay which is the mass is like some chemical potential force on charge okay and if you fix the charge here actually along this curve you get all the possible values of the charge so you don't realize that by doing that you're actually missing a huge part of the of the of the curves on a huge part of what could happen so that's it now second surprise do you have colors yes there is another solution so the schringer-deisen equation turned out to have two solutions not only the syk solution but another one which actually describes this perturbative regime oscillator harmonic like and this guy has zero entropy everywhere for all masses so you have two this guy has zero entropy it has also zero free energy this guy has some entropy and some free energy that you can compute and you can compare and you find there's a phase transition for some m which is not so i shouldn't call that mc let's call that m star there is no an mc here strictly less than m star where this solution becomes more stable than the green solution this mc is here and that's the first order phase transition because you jump from here to there now let me draw the entropy as function of mass so this was for zero temperature let me do it for some finite temperature you will have a curve here which is very similar do i have something to yeah that's okay that's okay i'm almost done okay you do it you find that the curve ends here not finite temperature doesn't go to zero anymore and the green solution still exists but also not for all masses it will do something like that so you still have a regime of parameters where actually two solutions coexist and you have to check out which one is the most stable and again you'll find some critical mass okay so that will be what we are starting to describe here is a line of first order phase transitions in this tn plane now i can keep going still increase the temperature and then the most remarkable feature of the model show up there's a critical point there's a critical point so that's now for t equal some very special temperature tc there's a critical point where this guy ends always like that and now this guy does exactly this and they meet here it means that this line of first order phase transition ends at a critical point which is very much like the gas liquid critical point in the phase diagram of water if you like and then if you go to higher temperature you have no longer two solutions you just have one solution to the Schringer-Daisen equation and this curve becomes a very boring curve monotonically decreasing curve like that which describes this region okay so this is the result this is the phase diagram of this of the model this region was was well known the rest is new and in particular so to come back maybe of what you on what you said that was one of the puzzle i had so here it's very very perturbative still if you turn on a little bit the temperature you expect indeed chaos you expect continuous a spectrum etc this is a finite temperature matrix quantum mechanics so it should have all these features so it's like a black hole here some sort of black hole very low entropy so it will be a super stringy small black hole whereas here it's much more like a holographic black hole large black hole with non-zero entropy so it can it can be extremal etc etc these two things look very very different and indeed in some sense there is a first order phase transition if you want to go from one to the other through this line here on the other hand they are not really different they are both chaotic they both have quasi normal behavior etc in the same sense as liquid water and and gas water are not really different and that's because you don't really have another parameter that distinguishes them and you can actually continuously join the two regions through this path in the parameter space if you like and so this is a way to continuously go from ordinary perturbation theory around the harmonic oscillator to this extremal black hole behavior with macroscopic non-zero entropy at zero temperature my last word is that okay that's interesting for fermions but that's vital for bosons because for bosons you don't have the perturbative regime here that allows you to do like syk if you want to understand systems with bosons you need to start always from a region like here this is the only region where you can define Feynman diagrams and that's why we were stuck for a lot of time in trying to understand the bosonic models because we thought they would be very similar to fermions but no they're actually not because they don't have the syk like perturbation theory and so you need to understand that you can have this very interesting sort of phase transitions in the in the parameter space